Bulk universality for generalized Wigner matrices

Abstract

Consider \(N × N\) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure \(\nu_{ij}\) with a subexponential decay. Let \(\sigma_{ij}^2\) be the variance for the probability measure \(\nu_{ij}\) with the normalization property that \(\sum_i\sigma_{ij}^2 = 1\) for all j. Under essentially the only condition that \(c\leq N\sigma_{ij}^2 \leq c^{−1}\) for some constant \(c > 0\), we prove that, in the limit \(N \rightarrow \infty\), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale \(M^{−1}\).